The purpose of this demo is to take the students through a wave speed calculation using
two different ropes with different masses per unit length and also being able to vary the
tension on the ropes. Students will be able to see the difference μ and T have on wave
speed. Relevant info, as well as answers, will be provided below.
Relevant equations, and example values:
$$v=\sqrt\frac T \mu$$
$$F_{net} = ma$$
$$\mu_{thick} = 29.2~\frac g m$$
$$\mu_{thin} = 6.4~\frac g m$$
$$4~kg\Rightarrow T=40~N$$
$$w_{Reg103} = 17.8~m$$
We will start by calculating μ. One meter long pieces of each rope can be measured using
a mass scale, and a meter stick. Dividing the mass by the length should give the values
listed above. Once these values are known, we can calculate the tension in the rope by
placing weights on the end of the rope that is passing over the pulley. Using Newton's
Second Law, we know that the tension in the rope must balance the weight of the masses.
$$F_{net} = ma \Rightarrow F_{net} = T - mg$$
Since the masses aren't moving:
$$T - mg = m*0 \Rightarrow T = mg$$
Now that we have both the tension, and μ, we are ready to calculate the wave speed v.
$$v=\sqrt{\frac {mg} \mu}$$
Using example values:
$$v=\sqrt\frac {4*9.8} {0.0292} \Rightarrow v \approx 116 \frac m s $$
The speed can be checked via demonstration; using a stopwatch, time how long it takes for a pulse
(made by hitting the string with a meter stick at one end) to cross the room and return a number of times
(to reduce error, time ten or twenty round trips). The room width is given above. The speed is then the
total distance divided by the total time. This can be repeated for the two different ropes, or with different
tensions. Best practice would be to go through one rope measuring everything first, and then measure the
wave speed for the other rope and have the students predict the mass of one meter of the rope.
...
This same setup will be used for a standing wave demonstration in the future.